A397579 a(n) = (4*Fibonacci(n) + Fibonacci(n+1))*Fibonacci(n+1)^3.

1, 5, 48, 297, 2125, 14336, 98865, 676053, 4637872, 31777625, 217835421, 1492992000, 10233313633, 70139666597, 480745758000, 3295076959881, 22584802594285, 154798515980288, 1061004875294097, 7272235438219125, 49844643644791216, 341640268890524345, 2341637241690712893

Offset

0,2

Comments

a(n) is the number of ways to tile a cross with four arms, each of length n (not counting the center cell) with squares and dominos. To illustrate, here is a cross with arm length n=3:

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Links

Table of n, a(n) for n=0..22.

Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1).

Formula

a(n) = A022095(n)*Fibonacci(n+1)^3 = A022095(n)*A056570(n+1).

a(n) = (9*Fibonacci(4*n+2) + 7*Fibonacci(4*n+3) + 8*Lucas(2*n+1)*(-1)^n - 6)/25.

a(n) = Fibonacci(n+1)^2*(Fibonacci(2*n+2) + 2*Fibonacci(n+1)*Fibonacci(n)).

a(n) = 5*a(n-1) + 15*a(n-2) - 15*a(n-3) - 5*a(n-4) + a(n-5).

G.f.: (3*x^3-8*x^2-1)/((x-1)*(x^2+3*x+1)*(x^2-7*x+1)). - Alois P. Heinz, Jul 01 2026

Example

For n=3, here is one of the a(3) = 297 ways to tile it with squares and dominos:
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      |_|.

Mathematica

Table[Fibonacci[n + 1]^3 (4 Fibonacci[n] + Fibonacci[n + 1]), {n, 0, 30}]

Crossrefs

Cf. A000032, A000045, A022095, A056570.

Sequence in context: A212111 A359969 A215050 * A268785 A395819 A268736

Adjacent sequences: A397575 A397576 A397577 * A397580 A397582 A397588

Keyword

nonn,easy,new

Author

Greg Dresden and Chuzhe (Justin) Ji, Jul 01 2026

Status

approved